My, what large numbers you are

The least common multiple of \( 2^ {12} \times 3^{23} \times 5^{45} \), \( 2^{23} \times 3^{45} \times 5^{12} \) and \( 2^{45} \times 3^{12} \times 5^{23} \) can be written as \( 2^a \times 3^b \times 5^c \), where \(a, b\) and \(c\) are non-negative integers. What is the value of \(a+b+c\)?

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